God is NaN

How skepticism leads back to a God, pragmatist mysticism, and the nonsense of Bayesian metaphysics

The other day, a liberating thought fell into my head. I can reject naturalism and the supernatural at the same time, for the simple reason that the dichotomy is devoid of meaning. Refusing it seems to be the only sensible thing to do. And, as it turns out, the world regains a whole lot of color upon discarding the false sense of closure provided by naturalism.

To get to the heart of the matter, we must first acknowledge that concepts are fuzzy. The frayed nature of a concept’s edge is immaterial most of the time, however. Context might tuck fuzzy boundaries differently for all of us, but in the end, the regions of parameter space where we agree tend to dominate those where we don’t. And, if we are willing to spin out sub- and sister-concepts like there is no tomorrow, we can jointly construct a more precise concept and together verify our agreement. Sure, we agree that this object is art of type-411, but it is not art of type-655.

To zoom in further, let me steal an example from Chapman (inspired by this great post by Richard Ngo, which discusses this example too):

There is water in the refrigerator.

Is this true? Your instinct might be to look in the fridge and say no if you don’t find a jug of water. But an annoying person might disagree; after all, there is condensation on the fridge walls, and all food contains H2O. We all intuitively understand the issue here. There is water in the refrigerator is not a sharply defined proposition. But if we really want to, we can increase its sharpness. Alice and Bob can have a discussion and come to the agreement that the original sentence is taken to mean the following:

There is a connected volume of space inside the fridge greater than 50mL which has a mass fraction of H2O greater than 97%.

Perhaps they agree to call this statement Type-I-watery-fridge. With this specification, they can check the fridge and come to agreement. At least if they aren’t sticklers. Bob might know Einstein’s general relativity and point out that space is subtle—in general conditions it is completely ill-defined without calibrating space measurements to a clock. And Alice knows that due to quantum effects, no particle can be truly localized. So if we are unlucky and measure a H2O fraction of 97.0000000001%, then we might worry that Type-I-watery-fridge is in a superposition of true and false. Or, alternatively, if the fridge is hovering near a black hole, it might be neither true nor false. It’s simply ill-defined. Or, less judgmentally, underdefined.

The solution here is not to become a Bayesian fetishist. Bayesian probability still operates on sharply defined propositions, so it is nonsense to give a Bayesian probability for the fridge near the black hole being a Type-I-watery-fridge.

The reality is that we are never able to make precise well-defined propositions about the world whose truth or falsity is determinable. We always rest on a thick bedrock of unquantifiable systematic error that renders every statement underdefined. Yet, this knowledge does not bother us; by some great and awe-inspiring mystery, we find ourselves in a world where most of the time we can just ignore it. We shrug, forget about the issue and, by some miracle, achieve our goals. Ignoring the ill-definedness of everything works.

It works.

That is the only real justification for us to proceed and play our underdefined language games. They work, and so we build temples, cure diseases, compose symphonies, and travel to the moon.

At this point, you might be tempted to point out that mathematics and logics are not ill-defined. Sure, there are genuine True and False statements, given some choice of axioms. The problem is that we are lacking a sharp interface between the mathematical world and whatever it is that we are experiencing. There is no theorem saying anything about my experience. Theorems and propositions are trapped in the realm of pure mathematics. And while that realm somehow appears to be deeply intertwined with whatever it is we are experiencing, its reassuring sharpness cannot be transported to sharp propositions about experience. There is no canonical correspondence between mathematical statements and the world as we access it. This is extremely important. Whatever words you utter or thoughts you think, there is no universal canonical translation into mathematics. All we can do is make some decision about how some piece of language or thought should mapped onto a sharp mathematical proposition, make some computation in that platonic realm, and then finally make some decision on how to do the reverse translation in the end. But you can never prove that you made the right translation in and out of the realm of mathematics. The only way to justify it, is that it works.

Thus, a proof that Bayesianism or standard logic is the right approach to epistemology cannot exist. But, you can observe that people using Bayesian reasoning make a lot of money in the stock market. More so than other techniques.

That said, strange alternative scenarios are on offer. Imagine meeting a technologically and artistically superior alien civilization that is friendly. After sorting out communication, you gradually come to understand that rather than using the standard two-valued logic of {True, False}, they use trivalent logic; a logic where every statement is either Frue, Galsh, or Trulse. They go further than all but the most hardcore human mathematicians who study multivalued logic, however; rather than studying three-valued logic from within two-valued logic, they evaluate everything from the perspective of three-valued logic. Every proposition is either Frue, Galsh or Trulse, and that’s the end of it. It sounds like a dumb idea, but if their civilization finds correspondence rules for translating in and out of their tri-mathematical realm that enable them to build spaceships and produce sublime sounds, then what can I say to them? If I say their system is False, all they’ll do is retort that my system is Trulse.

But we haven’t met trivalent aliens. And for a wide-but-not-all-encompassing range of tasks, two-valent logic and its Bayesian upgrade is the best we have. So we trust it for those types of tasks. But we should not trust it blindly. Bayesian epistemology is only as good as your correspondence rules. The only reason we can communicate and roughly agree what statements about probability mean most of the time is that we have a large amount of implicit agreement about correspondence rules. The translation task is relatively canonical, and we can resort to being increasingly more precise if needed, leading to convergence on agreement with relatively high precision.

No point building Bayesian metaphysique

But then someone comes along and tries to do metaphysics using logic or Bayesian epistemology, and everything turns to a mush of nonsense dressed in fancy garb. The correspondence rules lose all grounding. We take some words and pick our favorite conversion into abstract mathematical symbols. Then we do some computation in the Platonic realm and pick out favorite conversion back to natural language. But these correspondence rules never get sharpened. There is no way for two parties to reduce the severity of the underdefinedness of the statements they are making. There is nothing we can test, nothing we can do to converge to anything. So why should I pick your correspondence rules over mine? I cannot even conquer you and burn your cities to prove that my correspondence rules work better, because none of the correspondence rules needed for metaphysics are close to those needed for conquest—i.e. the correspondence rules of physics. I might be flogging a dead horse here, but it’s really important: in the realm of metaphysics, nothing lets us check if our correspondence rules work. So while there might be some value in metaphysics, that value cannot be cashed out in logical proposition.

As an example, consider someone trying to make a Bayesian argument for God. They might ask what is the probability that God exists given that I exist:

P(God exists | I exist).

Bayesian reasoning must operate on logical propositions, so let’s decide that “I exist” maps to the abstract logical proposition A. Every logic proposition has a negation, so the negation ¬A is a thing. How do we translate ¬A back to language? Probably, we both agree that it translates back to “I do not exist”. Phew... But actually, what is that supposed to mean? If I do not exist, what is “I” referring to? Maybe I can try to say that “I = a certain finite window of allowable configurations of atoms”, but how do I know that this is anything close to the right thing? What is the invariant property of atoms that is a sufficient condition to constitute “I”, but that is nevertheless flexible enough that it allows us to reason over hypothetical configuration spaces involving both “I”, “God”, and “not God” at the same time? Wouldn’t the sufficient condition for “I” strongly depend on whether God exists or not?

Naturalism and supernaturalism

So now, finally, what does it even mean to be a naturalist? How do I ever, even in principle, conceive of something happening that makes me significantly more confident that the world is supernatural over natural, or vice versa?

First, the naturalist can always claim that the phenomenon observed is due to currently unknown laws, since the notion of a law is sufficiently flexible. Physical systems can have external driving forces that depend on time, and the same could in principle happen to the universe as a whole. You might call that bizarre, but an external driving force on the universe is in fact indistinguishable from time-dependent “constants” of nature. So would we distinguish the hand of God from a time-dependent law of nature? The theoretical physicist in me wants to complain that this breaks a cherished principle: that the laws of physics should not privilege a particular coordinate system. But this has only been tested in certain energy regimes, and even more importantly, in sufficiently low-complexity regimes. So, being dead serious here: what is the distinction between a deity’s hand and constants of nature depending on time?

Thus, the naturalist can always weasel out by saying that the miracle that just happened was the revelation of a more complicated law. So really, what content does the stance of naturalism have that isn’t captured in totality by the following statement?

The parts of the world that are tidy and simple enough that we can measure them with high fidelity, appear to be governed by simple and elegant laws.

I mean, this is a fact as profound as any, and it is a great mystery that this is true. But, there are parts of the world that are not tidy nor measurable with high fidelity. Whether they are law-bound conceivably just depends on what you define as a law. And even then, laws live in the space of mathematics. How confident are you that you can find correspondence rules for these messy phenomena?

Conversely, turning to the non-naturalist, what does claiming that the world is not natural add to anything? We can anyway only ever capture patches and fragments of reality imperfectly with mathematics. Reality appears too vast to fit into any comprehensive symbol system we have found so far. So this giant unknowable blob of mystery that we are nodding towards when we say nature, what does it mean to say there is more than it? Doesn’t our severely incomplete understanding already leave enough room?

And with that, we arrive at a little question. Is the following a True statement?

God exists.

Well, unless someone specifies what God means to such high specificity that it fizzles out in provincial banality, my whole point is that you should refuse to even try answering. We have no correspondence rules for converting these words into the realm where True and False belong. And wow, how liberating it is to be in this predicament.

The reality of God

There is this little koan by a pragmatist philosopher, whom exactly I cannot remember, that really struck me when I heard it. It goes as follows

God doesn’t exist, but God is real.

I’ve been meditating on it the last several days. Ultimately, I take it not as saying that “God exists” is False, but rather, that the statement is so underdefined that the word exist and God cannot be juxtaposed in a way where we have a sensible way to convert this to the realm of logic. As the mantra goes, there are no correspondence rules, so there is no portal to the domain of True and False. Or rather you can pick any portal you want and get your favorite number for the probability of God’s existence. In this predicament, the thing we must realize is that God is Not a Number. Logic has no teeth. Thus, we find ourselves in a strange predicament. When the atheist confidently declares that God does not exist, he is guilty of not being skeptical enough.

We might point out that the religious are in equally deep trouble when they claim God exists. But if God is real is still a meaningful statement, maybe they can fall back to that? That leads to an intimidating question. Is there a sensible conception of Real that makes this meaningful?

Well, what does Real mean? I don’t know, like literally everything, it is an underdefined piece of language. There is no ultimate answer, no true definition—only 8 billion fuzzy slightly distinct versions that seem to overlap when we settle for some finite precision and basic examples. Yet, we can try to grasp at outlines. Almost everyone who takes the time to understand the theory of quantum electrodynamics (QED)—the theory of photons and electrons—will say that it is real. Now, this theory makes a remarkable prediction. It lets us compute that the so-called anomalous magnetic moment of an electron equals:

where the numbers in the parenthesis are uncertainties. Do you know what we measured in the lab?

The two match to 1 part in 10 billion, and they fall within each other’s uncertainties. To my knowledge, this is the most precise prediction humanity has ever made. Yet, we know that QED is only approximate. At sufficiently high energies, it does not describe the world: the photon isn’t a valid concept anymore. So if you try to map the piece of language “QED is true” into a mathematical proposition A, together with some appropriate experimental results implying something about A, you should probably obtain that A=False. Yet, QED is real, because it works. I don’t think there is any other sensible way to adjudicate realness without concluding that all our scientific theories are unreal. With that, let’s try this out: real things are things that work.

I’m starting now to wonder if being religious is the rational path, if we understand rationality properly. As much as I admire many of the rationalists, rationality cannot be about bowing down at the altar of Bayes. Most aspects of my conscious experience, which is as real as anything gets for me, have no approximate map into the space of propositions. Finding myself in this predicament, I should instead accept that being rational is doing what works; taking seriously what’s real.

But what does it mean for something to work? Work for what, exactly? Often we lose the forest for the trees and point out that what works is science, because it lets us construct bridges, hospitals, and computers that we can program bass music with. I’m sympathetic, I’ve dedicated an enormous amount of time to science and bass music. But ultimately, these are just secondary endpoints. The primary endpoints are the conscious states we are seeking. The aforementioned fruits of science are ultimately tools to more reliably produce the experiences we hunger for; they are indirect measures strongly correlating with the capability of humans getting what they want from existence.

But there are other approaches to achieving our desired states, relying on forms of knowledge whose value cannot be demonstrated through the tidy and easily measurable secondary endpoints approachable with science. Forms of knowledge that take the shape of ineffable correlation in raw experience. Perhaps we discover a religious ritual that enables us to experience the great mystery more directly, in ways that feel more real than anything else we have ever experienced. Why is that experience less real than what is predictable by science? Ultimately,

I do science because it works,
what works is what realizes my goals,
and my highest goal is sacred experience.

Why would it be that the thing that works is real only if it takes a detour through the realm of mathematics? I don’t know what is real other than what works, and the methods of the monk reliably lets you dip your head into the great mysteries—the sages teach you something real. And, words are by and for humans; if one group labels that thing that the sages help you touch God, while the other says God doesn’t exist, that makes for an easy choice. As much as it in-fights over provincial irrelevancies, the members of the first group overlap on a central core—they point to something real and give it a name. The second group wants to discard the word God, and so you have to invent a new name for your experience, at the same time cutting yourself off from a long lineage of insight that has stood the test of time. We don’t have to do that. Skepticism led us away, but it is also what leads us back.